> The vast, vast majority of real numbers cannot be named, not even in principle. Their definitions would have to be infinitely long.
So what? That doesn't stop them from being named.
So what? That doesn't stop them from being named.
(For a more compelling argument, the equality of real numbers is uncomputable, so it must be impossible to name them. If we could name them usefully, then we could determine equality by asking which ones had different names.)
No, I'm not.
> Most people would consider you not to have named something if you're literally never allowed to stop speaking (or writing) the name.
What's the connection? Names do not completely describe their referent. They're names. I know someone named Ko. Given that name, what can you tell me about Ko?
You can "name" a number such as: "a positive number, that when multiplied by itself, the result is 2".
This is a finite statement that requires only a handful of 'bits' to represent, but it exactly and uniquely identifies the square root of two. If written with decimal digits, then the square root of two would require an infinite number of digits, but it can be defined ('named') with a finite number of bits.
It turns out that this is a rare property for real numbers. The vast majority do not have a shorthand name like this, only the infinitely long description exists.
AC says: if there is a set of non-empty sets y with length x, I can construct a new set z of length x by taking a member from each set in y.
How does being able to name the constituents z or any member of y matter to the logic of the axiom?
You might consider that in the existing Chinese writing system, 口 and 囗 are distinct symbols.
Edit : actually does not even work, courtesy of the Chaitin's numbers